You should have specified, which gauge transformations where introduced in your course. In physics, quite a large number of examples go with this name. So, I am going to try to guess.
In classical mechanics, were you study non-relativistic motion of a particle, in most cases there only symmetries, that are present for the equations of motion are "global", that is universal over space. For example, the free particle action is invariant under shifts in space $x(t)\to x(t)+x_0$ (for each particle), or shifts in time $x(t)\to x(t+t_0)$. Both are very useful - first leads to momentum conservation, second to energy conservation.
However, one situation in classical mechanics is rather peculiar - that is the case of motion of a particle in electromagnetic field. While it is rather easy to write the Newtons equation of motion for such a particle in terms of well known electric and magnetic fields $E$ and $B$
$$
m\ddot{\mathbf{x}} = q\mathbf{E}+q\dot{\mathbf{x}}\times\mathbf{B},
$$
it turns out that the task to write an action, leading to these equations, is rather hard. To do so, it is required to introduce the scalar and vector potentials, $\phi$ and $\mathbf{A}$, with the action
$$
S=\int dt \left({1\over2} \,m \,\dot{\mathbf{x}}^2-e\, \mathbf{A}(t, \mathbf{x})\cdot \dot{\mathbf{x}}-e \,A_0(t, \mathbf{x})\right).
$$
This would lead to the proper equations of motion, given $\mathbf{E}=-\nabla A_0$, $\mathbf{B}=-\nabla\times\mathbf{A}$ (for static fields). However, it turns that this is not a unique choice! Any functions $A_0'(x,t)=A_0+d\alpha(x,t)/dt$, $\mathbf{A}'(x,t)=\mathbf{A}+\nabla\alpha(x,t)$ will lead to exactly the same equations of motion for the particle. Here $\alpha(x,t)$ is an arbitrary function of time and space - "gauge" transformation.
For now, this is a mathematical peculiarity. One could just use only Newton's laws of motion, electric and magnetic field, and never bother about vector potential and least acction principle. At the end, only gauge invariant quantities, $\mathbf{E}$ and $\mathbf{B}$ are really measured and physical! However, it turns out, that the formulation in terms of $\mathbf{A}$ and $A^0$ is very deep and useful, though it has this funny "gauge invariance" feature. Really, what happens that in fact the statement, that only gauge invariant quantities are observed goes beyond $E$ and $B$. Unfortunately, you can not really see that in classical mechanics, but in quantum theory this leads to real physical effects. This is the Aharonov–Bohm effect. If in a space both $E=B=0$, it is still possible to have nonzero $\mathbf{A}$, if $\nabla\times\mathbf{A}=0$. However, one can make configurations that an integral over a noncontractable loop
$$
\oint \mathbf{A} d\mathbf{x} \neq 0
$$
is not zero, though the magnetic field is zero everywhere along the loop. This integral is actually equal to the magnetic flux through the loop (say, there is an infinitely long solenoid inside the loop). And in quantum mechanics this leads to an interference pattern between particles that never enter the region with magnetic field, but only travel in the space around the solenoid! Thus, the description in terms of the scalar and vector potential is more complete, than with the magnetic and electric fields, at the price of an extra complication of gauge invariance.
Later on, in the quantum field theory, it turns out that the gauge principle governs all the fundamental forces of nature, not only electromagnetism.
As a side note, there is one more gauge transformation that may appear in mechanics, that is the reparametrisation invariance in teh world line action for a relativistic particle (a one dimensional analogue of a better known Polyakov action for a string), but this is probably better to leave until the study of general relativity and string theory.